Roots of F4

The 48 root vectors of F4 can be found as the vertices of the 24-cell in two dual configurations:

24-cell vertices:

24 roots by (±1,±1,0,0), permuting coordinate positions

Dual 24-cell vertices:

8 roots by (±1, 0, 0, 0), permuting coordinate positions

16 roots by (±½, ±½, ±½, ±½).

Simple roots

One choice of simple roots for F4,, is given by the rows of the following matrix:

\begin{bmatrix}
0&1&-1&0\\
0&0&1&-1\\
0&0&0&1\\

1

&-

2

1

&-

2

1

&-

2

1

2

\\
\end{bmatrix}

F4 polynomial invariant

Just as O(n) is the group of automorphisms which keep the quadratic polynomials x2 + y2 + ... invariant, F4 is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables).

Representations

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are :

The 52-dimensional representation is the adjoint representation, and the 26-dimensional one is the trace-free part of the action of F4 on the exceptional Albert algebra of dimension 27.

There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The fundamental representations are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the Dynkin diagram in the order such that the double arrow points from the second to the third).